PUMPA - THE SMART LEARNING APP

AI system creates personalised training plan based on your mistakes

### Theory:

Steps to find the irrational numbers between two numbers:
Step 1: Express the provided number in decimal form.

Step 2: Keep in mind that we need to construct irrational numbers between the provided numbers.

Step 3: Within the possible range construct irrational numbers such that they are non-recurring and non-terminating.
Example:
Consider two numbers $$1/6$$ and $$2/3$$.

The decimal expressions of provided numbers are $$1/6 = 0.1666...$$ and $$2/3 = 0.6666...$$

Now we have to construct irrational between $$0.1666...$$ and $$0.6666...$$

Remember that irrationals are non-recurring and non-terminating.

We ourself can construct numbers with the given conditions.

Let the numbers are $$0.17070070007....$$, $$0.23003000300003...$$, $$0.35010011011101111...$$

In this way, we can construct an infinite number of irrationals between any two numbers.

The easiest way to express an irrational number between two rational numbers:
Step 1: Express the smaller rational number as a terminating decimal.

Step 2: Put $$01$$ after its last decimal.

Step 3: Then put $$001$$ then put $$0001$$ then put $$00001$$ … … … so on and on.

Step 4: Hence the resultant number is an irrational number between the two given rational numbers.
Example:
Consider the same two numbers $$1/6$$ and $$2/3$$.

The decimal expressions of provided numbers are $$1/6 = 0.1666...$$ and $$2/3 = 0.6666...$$

Now we have to construct irrational between $$0.1666...$$ and $$0.6666...$$

Let us round off the smallest rational number to three decimal places.

Thus, the smallest rational number is $$0.167$$.

Now add the number $$01$$, it becomes $$0.16701$$.

Then add $$001$$, $$0.16701001$$.

Now add $$0001$$, $$0.167010010001$$

Then add $$00001$$, $$0.16701001000100001$$.

Proceeding in the same way, the number becomes as follows:

$$0.16701001000100001000001...$$